Finding the parameters for Harmonic Oscillator solutions

[gabu]

Homework Statement

Using the Schrödinger equation find the parameter $\alpha$ of the Harmonic Oscillator solution $\Psi(x)=A x e^{-\alpha x^2}$

Homework Equations

$-\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x)$

$E=\hbar\omega(n+\frac{1}{2})$

The Attempt at a Solution

Using the Schrödinger equation we have arrive at

$-2\alpha (2x^2\alpha-3)+\frac{m^2\omega^2x^2}{\hbar^2} = \frac{2m}{\hbar^2}E$

If I make x=0 I obtain

$\alpha = \frac{m\omega}{2\hbar}$

using the information that the energy level of the oscillator is the same as the highest power in the solution, meaning $E=3\hbar\omega/2$.

Now, my problem with this solution is the need to make x=0 to arrive at it. I know that the equation holds for every x, so it is justifiable to consider the origin. The thing is, however, that shouldn't I be able to solve the equation without this assumption? Shouldn't it be independent of x?

Thank you very much.

 

[TSny]

Hello.

Using the Schrödinger equation we have arrive at

$-2\alpha (2x^2\alpha-3)+\frac{m^2\omega^2x^2}{\hbar^2} = \frac{2m}{\hbar^2}E$

This equation must be satisfied for all $x$. This can be true only for one specific value of $\alpha$ and one specific value of $E$.

Group together terms of like powers in $x$ to form a polynomial in $x$. Use the fact that if a polynomial in $x$ is equal to zero for all values of $x$ in some finite domain, then each coefficient in the polynomial must be zero.

 

[Apashanka]

Using the Schrödinger equation find the parameter αα\alpha of the Harmonic Oscillator solution Ψ(x)=Axe−αx2Ψ(x)=Axe−αx2\Psi(x)=A x e^{-\alpha x^2}

This is actually the first excited state of a 1-D quantum harmonic oscillator having energy 3/2ħω
Where α is mω/2ħ

 

[gabu]

Hello.

This equation must be satisfied for all $x$. This can be true only for one specific value of $\alpha$ and one specific value of $E$.

Group together terms of like powers in $x$ to form a polynomial in $x$. Use the fact that if a polynomial in $x$ is equal to zero for all values of $x$ in some finite domain, then each coefficient in the polynomial must be zero.

Oh, I can see now. Thank you very much.